Suppose that p is the smallest prime that divides |G|. Show that anysubgroup of index p in G is normal in G.

Q: Assume that (G, ) is a group, that geG, and that t is the smallest positive integer such that g' =…

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Q: Prove B = {1/n : n in Natural numbers} is a union of {0} and that it is compact.

A: A set B is said to be compact if every open cover has a finite subcover of B.

Q: 3. Let m = 15. Compute (d) for every divisor d of m, and check that Edm (d) = m. Repeat this…

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Q: Compute the Lyapunov exponent of the map xn+1 = (4xn) mod 1 with two-digit accuracy.

A: To compute: The Lyapunov exponent of the map xn+1=4xn mod 1 with two-digit accuracy.

Q: Prove that if every subgraph of G has a node of degree at most d, then G is (d + 1)-colorable.

A: We know that G is itself a sub graph of G, so G has maximum vertex degree d. A standard result in…

Q: Exercise 14.4.22. Let o be a permutation in S,. (a) Show that there exists an integer k > 1 such…

A: Given that α is a permutation in Sn.

Q: 12. Suppose that k is a divisor of n. Prove that Z,/(k)~ Zp.

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Q: how that in D4 is not a normal subgroup. Please explain in details and show all the works.

A: To Show, a subgroup in D4 is not normal.

Q: Let m ≥ 2 be an integer. If [a]m does not = [b]m Show that [a]m ∩ [b]m = Ø.

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Q: Let n > 2 be an integer. Prove that An is a normal subgroup of Sn.

A: In abstract algebra, a normal subgroup is a subgroup that is invariant under conjugation by members…

Q: Exercise 8.8. Let p be a prime. Prove that ZpZp has exactly p+1 subgroups of order p.

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Q: Find the distinct equivalence classes for square-equivalence modulo m = 3, and observe that the…

A: To understand square-equivalence modulo m, we say that two integers a and b are square-equivalent…

Q: If N and K are submodules of M. (i) The kernel of the natural homomorphism N--> M/K is (N+K)/K. (ii)…

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Q: Let n be an integer greater than two. Show that no subgroup of order two is normal in Sn.

A: To prove that no subgroup of order 2 in the symmetric group Sn (n >2) is normal.

Q: 11.7 Let (rn) be an enumeration of the set Q of all rational numbers. Show there exists a…

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Q: Prove that the order of a k-cycle in S, is k.

A: Suppose σ is a k-cycle in the group Sn. then we follow that σ is a permutation of length k in which…

Q: 5.6.35. Prove that the set of all finite subsets of N is a countable set. Hint: For each k = NU {0},…

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Q: et n ≥ 3. (a) Find the number of permutations w ∈ Sn such that 1, 2, 3 belong to three distinct…

A: denotes symmetric group.

Q: Prove that the only nontrivial proper normal subgroup of S5 is A5.

A: We know that S5 is a group of all permutations on a set of five elements and also A5 is alternating…

Q: Construct a field F of cardinality 16 and find a generator of the group F∗.

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